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 A100058 Expansion of 1 / (1 - 3x - x^2 + 2x^3). 3
 1, 3, 10, 31, 97, 302, 941, 2931, 9130, 28439, 88585, 275934, 859509, 2677291, 8339514, 25976815, 80915377, 252043918, 785093501, 2445493667, 7617486666, 23727766663, 73909799321, 230222191294, 717120839877, 2233765112283 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n)/a(n-1) tends to 3.1149075414..., which is an eigenvalue of the matrix M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2. REFERENCES Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications", Fibonacci Association, 1993, p. 27. LINKS Index entries for linear recurrences with constant coefficients, signature (3, 1, -2). FORMULA Recurrence: a(0) = 1, a(1) = 3, a(2) = 10; a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3). Given Hosoya's triangle: 1; 1, 1; 2, 1, 2; considered as an upper triangular 3 X 3 matrix M: [2 1 2 / 1 1 0 / 1 0 0]; a(n) = center term in M^n * [1 0 0]. EXAMPLE a(5) = 97, center term in M^5 * [1 0 0]: [205 97 66]. MATHEMATICA CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *) Table[(MatrixPower[{{2, 1, 2}, {1, 1, 0}, {1, 0, 0}}, n].{1, 0, 0})[[2]], {n, 26}] (* Robert G. Wilson v, Nov 04 2004 *) LinearRecurrence[{3, 1, -2}, {1, 3, 10}, 30] (* Harvey P. Dale, Mar 28 2012 *) PROG (PARI) Vec(1/(1-3*x-x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012 CROSSREFS Partial sums of A052911. Cf. A019481, A052550, A052939, A100059, A058071. Sequence in context: A055217 A097472 A068094 * A002160 A214839 A114487 Adjacent sequences: A100055 A100056 A100057 * A100059 A100060 A100061 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Oct 31 2004 EXTENSIONS Edited by Ralf Stephan, Nov 02 2004 Corrected and extended by Robert G. Wilson v, Nov 04 2004 STATUS approved

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Last modified December 8 07:38 EST 2022. Contains 358691 sequences. (Running on oeis4.)